What kind of education is required for researchers in AI?
As a full-time AI researcher myself, I'd say that a PhD in machine learning would certainly be one useful option.

However, in order make much-needed progress, AI needs avoid falling into the trap of thinking that currently fashionable methods are any kind of 'silver bullet'. There's some danger that a PhD that heads straight into (say) some sub-sub-sub area of DL would end up imposing too much bias on the student's subsequent perspective.

AI research is an essentially multi-disciplinary activity. Other possible backgrounds therefore include:


Mathematics or physics (to first degree or PhD level). A strong background in either of these never did anyone any harm. People who are competent in these fields tend to be able to turn their abilities to new domains relatively easily. 
Software Engineering. One of the things that AI needs are integrative architectures for knowledge engineering. Here's why. I believe that one of the reasons that we haven't yet managed to do OCR at the level of a 5 year old is that we've yet to accept that we have to 'build a sledgehammer to crack a nut'. Software architects are used to managing large-scale complexity, so they may be able to help.
Cognitive Science, Psychology, Cognitive Linguistics. The reasons here are obvious.


Above all, I personally think that a good AI researcher should be creative, inquisitive and prepared to question received wisdom, all of which are more important in practice than specifics of their background.